Question Type 3: Related rates with geometric constraints Bootcamps

The radius of a right circular cylinder of fixed height $10\, cm$ is increasing at $0.1\, cm/s$. How fast is the volume increasing when the radius is 5 cm?

A spherical snowball melts so that its radius decreases at 0.2 cm/min. At what rate is its volume changing when the radius is 15 cm?

The area of a circle is increasing at $20 cm^2/s$. How fast is its radius increasing when the radius is 4 cm?

A ladder $100\, m$ long leans against a vertical wall. The bottom is pulled away from the wall at $5\, m/s$. How fast is the top sliding down the wall when the bottom is $64\, m$ from the wall?

In a right triangle one leg is fixed at 10 m while the other leg is growing at 0.4 m/s. How fast is the hypotenuse changing when the growing leg is 6 m?

A square has side length increasing at 0.5 cm/s. How fast is its diagonal increasing when the side length is 8 cm?

A $10$ ft ladder leans against a wall so that the angle $\theta$ with the ground changes as the base slides outward at $0.8 ft/s$. How fast is $\theta$ changing when the base is $6$ ft from the wall?

Water is being poured into a conical tank of height 12 m and base radius 5 m. The water level rises at 0.3 m/min. How fast is the volume of water increasing when the depth is 4 m?

A plane flying horizontally at 200 m/s passes directly over a radar station at altitude 3 km. How fast is the distance from the plane to the station increasing 30 s after it passes over?

An inverted conical tank (height 10 m, base radius 3 m) is leaking water at 2 m^3/min. How fast is the water depth decreasing when the water is 5 m deep?

Water flows into a conical vessel at 5 L/min. The vessel has height 4 m and base radius 2 m. How fast is the water level rising when the depth is 2 m? (1 L=0.001 m^3)

A balloon is rising vertically at $4 m/s$. A cyclist rides along a straight road at $12 m/s$ directly under the balloon's launch point. How fast is the distance between the cyclist and the balloon changing when the cyclist is $50 m$ from the launch point and the balloon is $80 m$ high?